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Properties

Label 2-63-63.25-c3-0-9
Degree $2$
Conductor $63$
Sign $-0.120 + 0.992i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.07·2-s + (−4.95 + 1.57i)3-s + 17.8·4-s + (−9.23 + 15.9i)5-s + (25.1 − 7.97i)6-s + (−8.82 + 16.2i)7-s − 49.8·8-s + (22.0 − 15.5i)9-s + (46.9 − 81.2i)10-s + (−8.59 − 14.8i)11-s + (−88.1 + 27.9i)12-s + (−27.2 − 47.2i)13-s + (44.8 − 82.7i)14-s + (20.6 − 93.7i)15-s + 110.·16-s + (−7.61 + 13.1i)17-s + ⋯
L(s)  = 1  − 1.79·2-s + (−0.953 + 0.302i)3-s + 2.22·4-s + (−0.826 + 1.43i)5-s + (1.71 − 0.542i)6-s + (−0.476 + 0.879i)7-s − 2.20·8-s + (0.817 − 0.576i)9-s + (1.48 − 2.56i)10-s + (−0.235 − 0.407i)11-s + (−2.12 + 0.672i)12-s + (−0.582 − 1.00i)13-s + (0.855 − 1.57i)14-s + (0.355 − 1.61i)15-s + 1.72·16-s + (−0.108 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.120 + 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0228368 - 0.0257758i\)
\(L(\frac12)\) \(\approx\) \(0.0228368 - 0.0257758i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.95 - 1.57i)T \)
7 \( 1 + (8.82 - 16.2i)T \)
good2 \( 1 + 5.07T + 8T^{2} \)
5 \( 1 + (9.23 - 15.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (8.59 + 14.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (27.2 + 47.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (7.61 - 13.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43.7 - 75.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48.3 + 83.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-57.3 + 99.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 50.0T + 2.97e4T^{2} \)
37 \( 1 + (-3.58 - 6.20i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (87.4 + 151. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (77.4 - 134. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 + (1.27 - 2.20i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 412.T + 2.05e5T^{2} \)
61 \( 1 + 711.T + 2.26e5T^{2} \)
67 \( 1 - 37.7T + 3.00e5T^{2} \)
71 \( 1 + 290.T + 3.57e5T^{2} \)
73 \( 1 + (-209. + 363. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 + (634. - 1.09e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-5.74 - 9.95i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-935. + 1.61e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.96710326705292718624688792396, −12.31016149689568942698098700308, −11.43554254332178204929249092600, −10.52717577239605577908253965728, −9.851294144592670359900253084956, −8.220890373520804065239028183453, −7.09059396729115486523293139062, −6.01811175799259449353724328410, −2.96329687332483962664842234267, −0.05371781253865702818002824200, 1.19269709162019315553369286858, 4.74613838395635553509503184119, 6.90126705109568542423537295578, 7.62452350936252533338351046449, 9.019596458567475597810252004360, 9.992510038347062509050022486929, 11.30053848948412588603678484542, 12.03535869725374500838615276378, 13.21674916012863622338302292988, 15.67905698696308119593927035850

Graph of the $Z$-function along the critical line